I have a Kenmore Refridgerator that I just moved. I had been sitting for a few months.

Duringt the move, we had to lay it on it's side for a few miles.

The freezer begun cooling right away when plugged in; however, the rest of the fridge has not suffered the same fate. I hear it takes about 24 hours for that to take place. Why would the freezer work right away and could there be an issue with the fridge?

Welcome to the Forum, Tony:
The freezer is where the cooling coil is located and the refrigerator simply circulates a measured amount of air between it and the freezer. So, if the freezer works, there is no reason the refrigerator would not.
Glenn